July  2011, 16(1): 1-14. doi: 10.3934/dcdsb.2011.16.1

The Euler-Maruyama approximations for the CEV model

1. 

School of Mathematical Sciences, Monash University, Clayton Campus, Building 28, Wellington road, Victoria, 3800, Australia

2. 

School of Mathematical Sciences, Monash University, Clayton Campus, Building 28,, Wellington road, Victoria, 3800, Australia

3. 

Department of Engineering Systems, Tel Aviv University, Tel Aviv, Ramat Aviv, 69978, Israel

Received  April 2010 Revised  August 2010 Published  April 2011

The CEV model is given by the stochastic differential equation $X_t=X_0+\int_0^t\mu X_s ds+\int_0^t\sigma (X^+_s)^p dW_s$, $\frac{1}{2}\le p<1$. It features a non-Lipschitz diffusion coefficient and gets absorbed at zero with a positive probability. We show the weak convergence of Euler-Maruyama approximations $X_t^n$ to the process $X_t$, $0 \le t \le T$, in the Skorokhod metric, by giving a new approximation by continuous processes. We calculate ruin probabilities as an example of such approximation. The ruin probability evaluated by simulations is not guaranteed to converge to the theoretical one, because the limiting distribution is discontinuous at zero. To approximate the size of the jump at zero we use the Levy metric, and also confirm the convergence numerically.
Citation: Vyacheslav M. Abramov, Fima C. Klebaner, Robert Sh. Lipster. The Euler-Maruyama approximations for the CEV model. Discrete and Continuous Dynamical Systems - B, 2011, 16 (1) : 1-14. doi: 10.3934/dcdsb.2011.16.1
References:
[1]

A. Alfonsi, Higher order discretization schemes for the CIR process: application to affine term structure and Heston models, Mathematics of Computation, 79 (2010), 209-237. doi: 10.1090/S0025-5718-09-02252-2.

[2]

V. Bally and D. Talay, The law of the Euler scheme for stochastic differential equations I. Convergence rate of the distribution function, Probability Theory and Related Fields, 104 (1996), 43-60. doi: 10.1007/BF01303802.

[3]

P. Billingsley, "Converges of Probability Measures," John Wiley & Sons, New York, 1968.

[4]

M. Bossy and A. Diop, An efficient discretisation scheme for one dimensional SDEs with a diffusion coefficient function of the form $|x|^\alpha, \alpha\in [1/2, 1)$, preprint, version 2, INRIA, France, 2007.

[5]

J. C. Cox, The constant elasticity of variance option. Pricing model, The Journal of Portfolio Management, 23 (1997), 15-17.

[6]

D. Dawson, http://www.math.ubc.ca/ db5d/SummerSchool09/lectures-dd/lecture4.pdf, (Accessed on August 3, 2010.)

[7]

G. Deelstra and F. Delbaen, Convergence of discretized stochastic (interest rate) processes with stochastic drift term, Applied Stochastic Models and Data Analysis, 14 (1998), 77-84. doi: 10.1002/(SICI)1099-0747(199803)14:1<77::AID-ASM338>3.0.CO;2-2.

[8]

F. Delbaen and H. A. Shirakawa, Note of option pricing for constant elasticity of variance model, Asia-Pacific Financial Markets, 9 (2002), 159-168. doi: 10.1023/A:1024173029378.

[9]

W. Feller, Two singular diffusion problems, Annals of Mathematics, 54 (1951), 173-182. doi: 10.2307/1969318.

[10]

I. Gy, A note on Eulers approximations, Potential Analysis, 8 (1998), 205-216. doi: 10.1023/A:1008605221617.

[11]

I. Gy,I. ongy and N. Krylov, Existence of strong solutions for Itó's stochastic equations via approximations, Probability Theory and Related Fields, 105 (1996), 143-158. doi: 10.1007/BF01203833.

[12]

N. Halidias and P. Kloeden, A note on the Euler-Maruyama scheme for stochastic differential equations with a discontinuous monotone drift coefficient, BIT Numerical Mathematics, 48 (2008), 51-59 . doi: 10.1007/s10543-008-0164-1.

[13]

P. L. Hennequin and A. Tortrat, "Théorie des Probabilités et Quelques Applications," Masson et Cie, Éditeurs, Paris, 1965.

[14]

D. J. Higham and X. Mao, Convergence of Monte Carlo simulations involving the mean-reverting square root process, Computational Finance, 8 (2005), 35-61.

[15]

D. J. Higham, X. Mao and A. M. Stuart, Strong convergence of Euler-type methods for nonlinear stochastic differential equations, SIAM Journal of Numerical Analysis, 40 (2002), 1041-1063. doi: 10.1137/S0036142901389530.

[16]

M. Hutzenthaler and A. Jentzen, Non-globally Lipschitz counterexamples for the stochastic Euler scheme, preprint, arXiv:0905.0273.

[17]

A. Jentzen, P. E. Kloeden and A. Neuenkirch, Pathwise approximation of stochastic differential equations on domains: Higher order convergence rates without global Lipschitz coeffcients, Numerische Mathematik, 112 (2009), 41-64. doi: 10.1007/s00211-008-0200-8.

[18]

Yu. Kabanov, R. Liptser and A. N. Shiryaev, Estimates of closeness in variation of probability measures (Russian), Dokl. Akad. Nauk SSSR, 278 (1984), 265-268.

[19]

F. C. Klebaner, "Introduction to Stochastic Calculus with Applications," 2nd edition, Imperial College Press, London, 2005.

[20]

P. E. Kloeden and K. Platten, "Numerical Solution of Stochastic Differential Equations," Springer-Verlag, Berlin, 1992.

[21]

R. Sh. Liptser and A. N. Shiryayev, "Theory of Martingales" [Translated from the Russian by K. Dzjaparidze] Mathematics and its Applications (Soviet Series), 49 Kluwer Academic Publishers Group, Dordrecht, 1989.

[22]

G. N. Milstein and M. V. Tretyakov, "Stochastic Numerics for Mathematical Physics," Springer-Verlag, Berlin, 2004.

[23]

L. C. G. Rogers and D. Williams, "Diffusions, Markov Processes, and Martingales. Vol. 2. Itô Calculus," Reprint of the 2nd edition, Cambridge University Press, Cambridge, 2000.

[24]

T. Shiga and S. Watanabe, Bessel diffusions as a one parameter family of diffusion processes, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 27 (1973), 37-46.

[25]

D. Talay, Simulation and numerical analysis of stochastic differential systems: A review, in "Probabilistic Methods in Applied Physics. Lecture Notes in Physics," 451, 63-106, Springer, Berlin, 1995.

[26]

L. Yan, The Euler scheme with irregular coefficients, Annals of Probability, 30 (2002), 1172-1194. doi: 10.1214/aop/1029867124.

[27]

H. Zähle, Weak approximation of SDEs by discrete time processes, Journal of Applied Mathematics and Stochastic Analysis, 2008 (2008), Article ID 275747, 15 pages.

show all references

References:
[1]

A. Alfonsi, Higher order discretization schemes for the CIR process: application to affine term structure and Heston models, Mathematics of Computation, 79 (2010), 209-237. doi: 10.1090/S0025-5718-09-02252-2.

[2]

V. Bally and D. Talay, The law of the Euler scheme for stochastic differential equations I. Convergence rate of the distribution function, Probability Theory and Related Fields, 104 (1996), 43-60. doi: 10.1007/BF01303802.

[3]

P. Billingsley, "Converges of Probability Measures," John Wiley & Sons, New York, 1968.

[4]

M. Bossy and A. Diop, An efficient discretisation scheme for one dimensional SDEs with a diffusion coefficient function of the form $|x|^\alpha, \alpha\in [1/2, 1)$, preprint, version 2, INRIA, France, 2007.

[5]

J. C. Cox, The constant elasticity of variance option. Pricing model, The Journal of Portfolio Management, 23 (1997), 15-17.

[6]

D. Dawson, http://www.math.ubc.ca/ db5d/SummerSchool09/lectures-dd/lecture4.pdf, (Accessed on August 3, 2010.)

[7]

G. Deelstra and F. Delbaen, Convergence of discretized stochastic (interest rate) processes with stochastic drift term, Applied Stochastic Models and Data Analysis, 14 (1998), 77-84. doi: 10.1002/(SICI)1099-0747(199803)14:1<77::AID-ASM338>3.0.CO;2-2.

[8]

F. Delbaen and H. A. Shirakawa, Note of option pricing for constant elasticity of variance model, Asia-Pacific Financial Markets, 9 (2002), 159-168. doi: 10.1023/A:1024173029378.

[9]

W. Feller, Two singular diffusion problems, Annals of Mathematics, 54 (1951), 173-182. doi: 10.2307/1969318.

[10]

I. Gy, A note on Eulers approximations, Potential Analysis, 8 (1998), 205-216. doi: 10.1023/A:1008605221617.

[11]

I. Gy,I. ongy and N. Krylov, Existence of strong solutions for Itó's stochastic equations via approximations, Probability Theory and Related Fields, 105 (1996), 143-158. doi: 10.1007/BF01203833.

[12]

N. Halidias and P. Kloeden, A note on the Euler-Maruyama scheme for stochastic differential equations with a discontinuous monotone drift coefficient, BIT Numerical Mathematics, 48 (2008), 51-59 . doi: 10.1007/s10543-008-0164-1.

[13]

P. L. Hennequin and A. Tortrat, "Théorie des Probabilités et Quelques Applications," Masson et Cie, Éditeurs, Paris, 1965.

[14]

D. J. Higham and X. Mao, Convergence of Monte Carlo simulations involving the mean-reverting square root process, Computational Finance, 8 (2005), 35-61.

[15]

D. J. Higham, X. Mao and A. M. Stuart, Strong convergence of Euler-type methods for nonlinear stochastic differential equations, SIAM Journal of Numerical Analysis, 40 (2002), 1041-1063. doi: 10.1137/S0036142901389530.

[16]

M. Hutzenthaler and A. Jentzen, Non-globally Lipschitz counterexamples for the stochastic Euler scheme, preprint, arXiv:0905.0273.

[17]

A. Jentzen, P. E. Kloeden and A. Neuenkirch, Pathwise approximation of stochastic differential equations on domains: Higher order convergence rates without global Lipschitz coeffcients, Numerische Mathematik, 112 (2009), 41-64. doi: 10.1007/s00211-008-0200-8.

[18]

Yu. Kabanov, R. Liptser and A. N. Shiryaev, Estimates of closeness in variation of probability measures (Russian), Dokl. Akad. Nauk SSSR, 278 (1984), 265-268.

[19]

F. C. Klebaner, "Introduction to Stochastic Calculus with Applications," 2nd edition, Imperial College Press, London, 2005.

[20]

P. E. Kloeden and K. Platten, "Numerical Solution of Stochastic Differential Equations," Springer-Verlag, Berlin, 1992.

[21]

R. Sh. Liptser and A. N. Shiryayev, "Theory of Martingales" [Translated from the Russian by K. Dzjaparidze] Mathematics and its Applications (Soviet Series), 49 Kluwer Academic Publishers Group, Dordrecht, 1989.

[22]

G. N. Milstein and M. V. Tretyakov, "Stochastic Numerics for Mathematical Physics," Springer-Verlag, Berlin, 2004.

[23]

L. C. G. Rogers and D. Williams, "Diffusions, Markov Processes, and Martingales. Vol. 2. Itô Calculus," Reprint of the 2nd edition, Cambridge University Press, Cambridge, 2000.

[24]

T. Shiga and S. Watanabe, Bessel diffusions as a one parameter family of diffusion processes, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 27 (1973), 37-46.

[25]

D. Talay, Simulation and numerical analysis of stochastic differential systems: A review, in "Probabilistic Methods in Applied Physics. Lecture Notes in Physics," 451, 63-106, Springer, Berlin, 1995.

[26]

L. Yan, The Euler scheme with irregular coefficients, Annals of Probability, 30 (2002), 1172-1194. doi: 10.1214/aop/1029867124.

[27]

H. Zähle, Weak approximation of SDEs by discrete time processes, Journal of Applied Mathematics and Stochastic Analysis, 2008 (2008), Article ID 275747, 15 pages.

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